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On the Complexity of Intersecting Regular, Context-Free, and Tree Languages

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 9135)

Abstract

We apply a construction of Cook (1971) to show that the intersection non-emptiness problem for one PDA (pushdown automaton) and a finite list of DFA’s (deterministic finite automata) characterizes the complexity class P. In particular, we show that there exist constants \(c_1\) and \(c_2\) such that for every k, intersection non-emptiness for one PDA and k DFA’s is solvable in \(O(n^{c_1 k})\) time, but is not solvable in \(O(n^{c_2 k})\) time. Then, for every k, we reduce intersection non-emptiness for one PDA and \(2^k\) DFA’s to non-emptiness for multi-stack pushdown automata with k-phase switches to obtain a tight time complexity lower bound. Further, we revisit a construction of Veanes (1997) to show that the intersection non-emptiness problem for tree automata also characterizes the complexity class P. We show that there exist constants \(c_1\) and \(c_2\) such that for every k, intersection non-emptiness for k tree automata is solvable in \(O(n^{c_1 k})\) time, but is not solvable in \(O(n^{c_2 k})\) time.

Keywords

  • Turing Machine
  • Input String
  • Tree Automaton
  • Input Position
  • Acceptance Problem

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Correspondence to Michael Wehar .

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Swernofsky, J., Wehar, M. (2015). On the Complexity of Intersecting Regular, Context-Free, and Tree Languages. In: Halldórsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds) Automata, Languages, and Programming. ICALP 2015. Lecture Notes in Computer Science(), vol 9135. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47666-6_33

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  • DOI: https://doi.org/10.1007/978-3-662-47666-6_33

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